Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAM-modulus and Hausdorff measure of codimension one in metric measure spaces
83
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $Gamma(E)$ be the family of all paths which meet a set $E$ in the metric measure space $X$. The set function $E mapsto AM(Gamma(E))$ defines the $AM$--modulus measure in $X$ where $AM$ refers to the approximation modulus. We compare $AM(Gamma(E))$ to the Hausdorff measure $comathcal H^1(E)$ of codimension one in $X$ and show that $$comathcal H^1(E) approx AM(Gamma(E))$$ for Suslin sets $E$ in $X$. This leads to a new characterization of sets of finite perimeter in $X$ in terms of the $AM$--modulus. We also study the level sets of $BV$ functions and show that for a.e. $t$ these sets have finite $comathcal H^1$--measure. Most of the results are new also in $mathbb R^n$.
rate research
Read More
In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
We show that given a homeomorphism $f:GrightarrowOmega$ where $G$ is a open subset of $mathbb{R}^2$ and $Omega$ is a open subset of a $2$-Ahlfors regular metric measure space supporting a weak $(1,1)$-Poincare inequality, it holds $fin BV_{operatorname{loc}}(G,Omega)$ if and only $f^{-1}in BV_{operatorname{loc}}(Omega,G)$. Further if $f$ satisfies the Luzin N and N$^{-1}$ conditions then $fin W^{1,1}_{operatorname{loc}}(G,Omega)$ if and only if $f^{-1}in W^{1,1}_{operatorname{loc}}(Omega,G)$.
In this note we continue giving the characterisation of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. This is a continuation of our paper [M. Ruzhansky and D. Verma. Hardy inequalities on metric measure spaces, Proc. R. Soc. A., 475(2223):20180310, 2018] where we treated the case $pleq q$. Here the remaining range $p>q$ is considered, namely, $0<q<p$, $1<p<infty.$ We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. We note that doubling conditions are not required for our analysis.
We prove that on an essentially non-branching $mathrm{MCP}(K,N)$ space, if a geodesic ball has a volume lower bound and satisfies some additional geometric conditions, then in a smaller geodesic ball (in a quantified sense) we have an estimate on the isoperimetric constants.
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. This allows us to study the MDS embeddings of the geodesic circle $S^1$ into $mathbb{R}^m$ for all $m$, and to ask questions about the MDS embeddings of the geodesic $n$-spheres $S^n$ into $mathbb{R}^m$. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space $X$, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of $X$? Convergence is understood when each metric space in the sequence has the same finite number of points, or when each metric space has a finite number of points tending to infinity. We are also interested in notions of convergence when each metric space in the sequence has an arbitrary (possibly infinite) number of points.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
